3.579 \(\int \frac{A+B x^2}{x^4 \left (a+b x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=82 \[ \frac{2 b x (4 A b-3 a B)}{3 a^3 \sqrt{a+b x^2}}+\frac{4 A b-3 a B}{3 a^2 x \sqrt{a+b x^2}}-\frac{A}{3 a x^3 \sqrt{a+b x^2}} \]

[Out]

-A/(3*a*x^3*Sqrt[a + b*x^2]) + (4*A*b - 3*a*B)/(3*a^2*x*Sqrt[a + b*x^2]) + (2*b*
(4*A*b - 3*a*B)*x)/(3*a^3*Sqrt[a + b*x^2])

_______________________________________________________________________________________

Rubi [A]  time = 0.10774, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{2 b x (4 A b-3 a B)}{3 a^3 \sqrt{a+b x^2}}+\frac{4 A b-3 a B}{3 a^2 x \sqrt{a+b x^2}}-\frac{A}{3 a x^3 \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x^2)/(x^4*(a + b*x^2)^(3/2)),x]

[Out]

-A/(3*a*x^3*Sqrt[a + b*x^2]) + (4*A*b - 3*a*B)/(3*a^2*x*Sqrt[a + b*x^2]) + (2*b*
(4*A*b - 3*a*B)*x)/(3*a^3*Sqrt[a + b*x^2])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 11.4035, size = 75, normalized size = 0.91 \[ - \frac{A}{3 a x^{3} \sqrt{a + b x^{2}}} + \frac{4 A b - 3 B a}{3 a^{2} x \sqrt{a + b x^{2}}} + \frac{2 b x \left (4 A b - 3 B a\right )}{3 a^{3} \sqrt{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**2+A)/x**4/(b*x**2+a)**(3/2),x)

[Out]

-A/(3*a*x**3*sqrt(a + b*x**2)) + (4*A*b - 3*B*a)/(3*a**2*x*sqrt(a + b*x**2)) + 2
*b*x*(4*A*b - 3*B*a)/(3*a**3*sqrt(a + b*x**2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0626492, size = 61, normalized size = 0.74 \[ \frac{-a^2 \left (A+3 B x^2\right )+a \left (4 A b x^2-6 b B x^4\right )+8 A b^2 x^4}{3 a^3 x^3 \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x^2)/(x^4*(a + b*x^2)^(3/2)),x]

[Out]

(8*A*b^2*x^4 - a^2*(A + 3*B*x^2) + a*(4*A*b*x^2 - 6*b*B*x^4))/(3*a^3*x^3*Sqrt[a
+ b*x^2])

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 58, normalized size = 0.7 \[ -{\frac{-8\,A{b}^{2}{x}^{4}+6\,Bab{x}^{4}-4\,aAb{x}^{2}+3\,B{a}^{2}{x}^{2}+A{a}^{2}}{3\,{x}^{3}{a}^{3}}{\frac{1}{\sqrt{b{x}^{2}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)/x^4/(b*x^2+a)^(3/2),x)

[Out]

-1/3*(-8*A*b^2*x^4+6*B*a*b*x^4-4*A*a*b*x^2+3*B*a^2*x^2+A*a^2)/(b*x^2+a)^(1/2)/x^
3/a^3

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/((b*x^2 + a)^(3/2)*x^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.231173, size = 92, normalized size = 1.12 \[ -\frac{{\left (2 \,{\left (3 \, B a b - 4 \, A b^{2}\right )} x^{4} + A a^{2} +{\left (3 \, B a^{2} - 4 \, A a b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{3 \,{\left (a^{3} b x^{5} + a^{4} x^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/((b*x^2 + a)^(3/2)*x^4),x, algorithm="fricas")

[Out]

-1/3*(2*(3*B*a*b - 4*A*b^2)*x^4 + A*a^2 + (3*B*a^2 - 4*A*a*b)*x^2)*sqrt(b*x^2 +
a)/(a^3*b*x^5 + a^4*x^3)

_______________________________________________________________________________________

Sympy [A]  time = 25.6826, size = 284, normalized size = 3.46 \[ A \left (- \frac{a^{3} b^{\frac{9}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{3 a^{5} b^{4} x^{2} + 6 a^{4} b^{5} x^{4} + 3 a^{3} b^{6} x^{6}} + \frac{3 a^{2} b^{\frac{11}{2}} x^{2} \sqrt{\frac{a}{b x^{2}} + 1}}{3 a^{5} b^{4} x^{2} + 6 a^{4} b^{5} x^{4} + 3 a^{3} b^{6} x^{6}} + \frac{12 a b^{\frac{13}{2}} x^{4} \sqrt{\frac{a}{b x^{2}} + 1}}{3 a^{5} b^{4} x^{2} + 6 a^{4} b^{5} x^{4} + 3 a^{3} b^{6} x^{6}} + \frac{8 b^{\frac{15}{2}} x^{6} \sqrt{\frac{a}{b x^{2}} + 1}}{3 a^{5} b^{4} x^{2} + 6 a^{4} b^{5} x^{4} + 3 a^{3} b^{6} x^{6}}\right ) + B \left (- \frac{1}{a \sqrt{b} x^{2} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{2 \sqrt{b}}{a^{2} \sqrt{\frac{a}{b x^{2}} + 1}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x**2+A)/x**4/(b*x**2+a)**(3/2),x)

[Out]

A*(-a**3*b**(9/2)*sqrt(a/(b*x**2) + 1)/(3*a**5*b**4*x**2 + 6*a**4*b**5*x**4 + 3*
a**3*b**6*x**6) + 3*a**2*b**(11/2)*x**2*sqrt(a/(b*x**2) + 1)/(3*a**5*b**4*x**2 +
6*a**4*b**5*x**4 + 3*a**3*b**6*x**6) + 12*a*b**(13/2)*x**4*sqrt(a/(b*x**2) + 1)
/(3*a**5*b**4*x**2 + 6*a**4*b**5*x**4 + 3*a**3*b**6*x**6) + 8*b**(15/2)*x**6*sqr
t(a/(b*x**2) + 1)/(3*a**5*b**4*x**2 + 6*a**4*b**5*x**4 + 3*a**3*b**6*x**6)) + B*
(-1/(a*sqrt(b)*x**2*sqrt(a/(b*x**2) + 1)) - 2*sqrt(b)/(a**2*sqrt(a/(b*x**2) + 1)
))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.240637, size = 244, normalized size = 2.98 \[ -\frac{{\left (B a b - A b^{2}\right )} x}{\sqrt{b x^{2} + a} a^{3}} + \frac{2 \,{\left (3 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} B a \sqrt{b} - 3 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} A b^{\frac{3}{2}} - 6 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} B a^{2} \sqrt{b} + 12 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} A a b^{\frac{3}{2}} + 3 \, B a^{3} \sqrt{b} - 5 \, A a^{2} b^{\frac{3}{2}}\right )}}{3 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{3} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/((b*x^2 + a)^(3/2)*x^4),x, algorithm="giac")

[Out]

-(B*a*b - A*b^2)*x/(sqrt(b*x^2 + a)*a^3) + 2/3*(3*(sqrt(b)*x - sqrt(b*x^2 + a))^
4*B*a*sqrt(b) - 3*(sqrt(b)*x - sqrt(b*x^2 + a))^4*A*b^(3/2) - 6*(sqrt(b)*x - sqr
t(b*x^2 + a))^2*B*a^2*sqrt(b) + 12*(sqrt(b)*x - sqrt(b*x^2 + a))^2*A*a*b^(3/2) +
3*B*a^3*sqrt(b) - 5*A*a^2*b^(3/2))/(((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^3*a^2
)